The zero knowledge proof

6 December 2023

Chalkdust HQ has been infiltrated by a secret non-mathematician! So we may as well make them attempt some maths. This issue we asked our secret non-mathematician to…

Prove that you cannot square the circle!

Well firstly, I find it surprising to be asked to prove that I cannot square the circle. I can hardly calculate the area of a circle in the first place. I first encountered the theorem that the area of a square cannot equal that of a circle (or in the pithy words of my editor, one can’t “construct a square with the area of a given circle by using only a finite number of steps with a compass and straightedge”) in the Arctic Monkeys song Don’t Sit Down ‘Cause I’ve Moved Your Chair. The brilliant Alex Turner there points out the danger of trying to “fill in a circular hole with a peg that’s square”. I always knew the man was a genius, but I now realise his talents extend to geometry too.

Yet the question of why one of you cannot square the circle certainly does vex me… I thought mathematicians were supposed to be the smartest of the smart? Shouldn’t you have solved it by now? I recall that Thomas Hobbes—the famous political philosopher—got into hot water in 1665 for claiming that he could square the circle. This was not surprising from a man who was fond of saying that if he had read as many books as other people, he would be as stupid as they were (some mathematicians may relate). But mathematicians since then seem to have lost their ambition.

I know, I know, you say it’s ‘impossible’. But take this advice from a former politics student: one should never let something as trifling as an impossibility get in the way of a good proof. It seems you are no strangers to this kind of thing, anyway. Do you really mean to tell me that imaginary numbers are ‘possible’? Put some of them to work on this squaring the circle business and you might meet with success.

Sadly for you, I have got there first. To square the circle you just need an infinitely large circle and infinitely large square. There, boom, you have it. Both areas are infinitely large so it’s problem solved. “That’s cheating!” I hear you saying, “you can’t use infinities!” Oh really, can’t I? Then show me a circle that actually has one side. Circles are not real shapes. There is no such thing as a circle. A curve is simply lots of small straight lines put together. Any low-resolution circle on MS Paint will show you this fact. So, far from being a one-sided shape, the circle is in fact an infinitely-sided shape. This means that if I am cheating by using infinity, then the circle is already cheating by being an infinitely-sided shape. Or (as is much more likely since infinities do not exist) the circle does not exist, and the question simply becomes how do you square the chiliagon (just Googled this) or some other such polygon with many sides. And that, dear reader, is one for you to solve. I think you will agree that I have done more than enough.

I was asked to prove that I can’t square the circle, and I am afraid I have shown the opposite. Be grateful; there is now one fewer problem for you to solve. And so we find ourselves back where we started; I suppose we have come full square.